# What is the shortest Ph.D. thesis? [closed]

The question is self-explanatory, but I want to make some remarks in order to prevent the responses from going off into undesirable directions.

It seems that every few years I hear someone ask this question; it seems to hold a perennial fascination for research mathematicians, just as quests for short proofs do. The trouble is that it has strong urban-legend tendencies: someone will say, "So-and-so's thesis was only $\epsilon$ pages long!" where $\epsilon \ll 1$. It will often be very difficult to confirm or disconfirm such claims, since Ph.D. theses are often not even published, let alone readily available online. If you Google around for a while, as I did, you will find many dubious leads and can easily waste a lot of time on wild goose chases. Frankly, I'm a bit fed up with this state of affairs. I am therefore asking this question on MO in the hope that doing so will put this old question to rest, or at least establish provable upper bounds.

I would therefore request that you set yourself a high standard before replying. Don't post a candidate unless you're sure your facts are correct, and please give some indication why you're so sure. Read the meta discussion before posting. (Note that the meta discussion illustrates that even a MathSciNet citation isn't always totally definitive.) Include information about the content and circumstances of the thesis if you know it, but resist the temptation to gossip or speculate.

I'm not making this question community wiki or big-list because it should ideally have a definite answer, though I grant that it's possible that there are some borderline cases out there (perhaps there are theses that were not written in scholarly good faith, or documents that some people would regard as equivalent to a Ph.D. thesis but that others would not, or theses in subjects that are strictly speaking distinct from mathematics but that are arguably indistinguishable from mathematics dissertations).

Finally, to anticipate a possible follow-up question, there is a list of short published papers here (search for "Nelson"). Note that the question of the shortest published paper is not as urban-legendy because the facts are easier to verify. I looked up the short papers listed there myself and found them to be quite interesting. So in addition to trying to settle an urban legend, I am hoping that this question will bring to light some interesting and lesser known mathematics.

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## closed as off topic by Andres Caicedo, Tom Leinster, Loop Space, Mark Meckes, Ryan BudneyFeb 9 '11 at 16:01

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I think it really should be CW. It makes no sense to me that the shorter the proposed candidate, the more reputation the proposer will get. It will also lower the temptation for people to post gossipy stuff. –  Alex B. Feb 8 '11 at 15:31
The only reasonable interpretation of the question is extremely short theses in general, because there is more than one measure of the length of a thesis. Moreover in some cases it's debatable whether a particular document really is a thesis or the full thesis. It realy should be CW. –  Greg Kuperberg Feb 8 '11 at 15:40
-1. This question is terrible. I'm sure I could reformat my thesis in a silly font size to make it have a ludicrously small number of pages. –  Peter McNamara Feb 8 '11 at 19:50
@Peter McNamara: you probably could, but I'm pretty certain that this is not the issue being discussed here. Anyway, most universities have specific formatting standards and would not let you submit it in this form. –  Thierry Zell Feb 8 '11 at 20:05
–  Gunnar Þór Magnússon Feb 8 '11 at 22:08

David Rector's thesis ("An Unstable Adams Spectral Sequence", MIT 1966) is 9 pages, according to the record at the MIT library. I haven't seen the actual thesis for many years, but I'm pretty the actual mathematical content takes about 3 pages total, and is largely identical to the published version in Topology (1966, same title), which is 3 pages plus bibliography. (Dan Kan, his advisor, likes short papers.)

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I see Tyler already mentioned this one on meta. –  Charles Rezk Feb 8 '11 at 20:09
Probably not a coincidence. –  Tyler Lawson Feb 8 '11 at 20:25
Accepted provisionally. Enough people seem instinctively annoyed at this question that it seems likely to be closed soon (despite the fact that I'm asking it on MO in order to prevent its proliferation elsewhere). It doesn't seem likely that a stronger candidate will emerge before then. Ideally I'd like to examine the thesis myself before accepting the answer but I don't feel like purchasing it and it may be a while before my next trip to Boston. –  Timothy Chow Feb 9 '11 at 15:56
Aside from the library copy, there should be a slightly more accessible copy in the MIT Math reading room. (They used to keep copies of theses there, and I assume they still do.) Maybe somebody reading this could wander down the hall and take a look. :) –  Charles Rezk Feb 9 '11 at 18:55
I'm in the reading room now. Rector's thesis comprises a title page, an abstract page, a table of contents page, 7 pages of math, a bibliography page (8 refs.), and a biographical note page. The MIT library record's "9 leaves" exclude the title/abstract/contents, which are not numbered. Except for some trivial changes in wording in the intro, the mathematical part is indeed identical to the 4-page Topology paper, vol. 5 (1966), 343-346. The thesis occupies more space since it's manually typed; not including section titles, the 4 sections are respectively 18, 23, 42, and 36 typewritten lines. –  Timothy Chow Aug 19 '11 at 18:44

John Nash's thesis was 26 pages, and had two references in the bibliography.

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Edmund Landau's thesis was 13 pages long.

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There is an English translation here: arxiv.org/PS_cache/arxiv/pdf/0803/0803.3787v2.pdf That document is 17 pages (including title page, etc.). –  Zach N Feb 8 '11 at 18:06
For a link to a scanned version of Landau's thesis see here gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN317979566 The document has 18 pages, of which 2 are completely empty, indeed the catalogue of the libraries of Berlin gives 16 pages as lengths. (the French national library catalogue gives 18). Moreover, one page is a title page, one is a dedication, and one is a vita. So, depending on what one actually counts, 18, 16, or 13. According to library catalogues 16 or 18. –  quid Feb 8 '11 at 18:15

I believe the shortest PhD thesis is of Burt Totaro "Milnor K-theory is the simplest part of algebraic K-theory", 12 pages.

Milnor K-theory is the simplest part of algebraic K-theory, Ph.D. thesis, University of California, Berkeley, 1989; K-Theory 6 (1992), 177-189.

Burt Totaro's webpage at Cambridge

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But is that the complete thesis or the paper that resulted from it? –  Mariano Suárez-Alvarez Feb 8 '11 at 17:40
its complete thesis. I gave two references here, Milnor K-theory is the simplest part of algebraic K-theory, Ph.D. thesis, University of California, Berkeley, 1989 and K-Theory 6 (1992), 177-189 –  J Verma Feb 8 '11 at 17:43
I noticed, but the reference to the actual thesis does not have a page numbers (and it is somewhat surprising that the number of pages did not change from the thesis to K-theory's format) :) –  Mariano Suárez-Alvarez Feb 8 '11 at 17:45
Totaro's 1989 thesis is titled "K-theory and algebraic cycles" and, according to ProQuest, is 20 pages. If your university library subscribes to ProQuest, you can see a PDF preview of the thesis by searching for "Totaro, Burt" in the Dissertations and Theses database. –  Zach N Feb 8 '11 at 18:02
You can download it on mathscinet. It has 16 numbered pages, incl. 1 page of bibliography. Definitions start on page 1 though, not much of an introduction. –  fherzig Feb 9 '11 at 2:58

This is not really an answer because these PhD's were never actually written, but anyway: in his book A mathematicians miscellany (in the chapter on math with minimum raw material) Littlewood gave 2 examples that could have been 2-line PhDs:

(1) Cayley's projective definition of length

(2)Theorem: An integral function never 0 or 1 is a constant. Proof: $\exp(i\Omega(f(z)))$ is a bounded integral function. ($\Omega$ is inverse to the elliptic modular function.)

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Richard, perhaps you overlooked that Gerry Myerson already gave this example on the meta discussion? –  Timothy Chow Feb 8 '11 at 15:53
I don't think it is reasonable to expect people to have read all the meta discussion before posting on a regular thread. This is a sort of fluff question, so it doesn't matter much, but in general I think it should be fine to repost answers from meta, so that the main thread has the most complete record of answers to the question. –  David Speyer Feb 8 '11 at 16:52
While I agree with David Speyer in general, I also do not think this should have been posted as an answer to this particular question, given the questioner's emphasis on restricting the scope of the question. –  Charles Staats Feb 8 '11 at 17:27
@David: I too would agree that in general it’s not reasonable to expect people to read meta discussions on questions before answering them. But this question specifically asks us to, and gives good reasons for it. –  Peter LeFanu Lumsdaine Feb 8 '11 at 20:18

I already posted this on meta where there was some discussion of whether the page count was correct. My guess is that it is, so I will post it here too:

MR2615548 Martens, Henrik Herman Buvik A NEW PROOF OF TORELLI'S THEOREM. Thesis (Ph.D.)–New York University. 1962. 12 pp.

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Compared to that, the thesis of his student Kristian Seip was a massive tome, weighing in at 30 pages. –  Harald Hanche-Olsen Feb 9 '11 at 7:56

Kurt Gödel seems to be a good candidate for this "prize".

Let me quote from this review (see Page 74) of Kurt Gödel Collected Works.

The first three works of Godel in this volume are his dissertation of 1929 (twenty-one pages in English), a revised and substantially abbreviated version (eleven pages in English) published in 1930, and a brief abstract based on a presentation of Godel's results in Konigsberg on 6 September 1930. Of all of Godel's longer, published writings, his dissertation has been, until now, the most difficult to obtain, and is here translated for the first time into English, by Stefan Bauer-Mengelberg and van Heijenoort.

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The original version of his thesis seems to have 33 pages; see permalink.obvsg.at/AC05181322 (the number next to "Umfangsangabe") –  quid Feb 8 '11 at 16:59
I cannot say anything about the original version (my German skills are null, not almost null). But I have just checked my copy of the Collected Works (unfortunately I have not found any online library to link), and in pages 60-101 we can find Godel's dissertation (even pages match German, while odd ones match English). Thus, the description "21 pages in English" is accurate. –  boumol Feb 8 '11 at 17:14
I did not want to imply your claim was not accurate. Only, as I understand the question, it is about the actual document the person submitted as a thesis. Thus, I supplemented this information, documenting it by the link to the entry of Goedels thesis in the joint library catalogue of Austrian (academic) libraries. It specifies title, author, year, lengths (that's the Umfangsangabe, S. abbreviates 'Seiten' i.e. pages), the type of document (thesis of University of Vienna (Wien)), and finally the specific libraries where it can be found. –  quid Feb 8 '11 at 17:48

According to mathscinet, Eva Kallin's thesis was 14 pages.

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This is promising, but as the question mentions and the meta thread shows, MathSciNet alone is not an authoritative reference. More documentation? –  Peter LeFanu Lumsdaine Feb 8 '11 at 20:12

Barry Mazur's thesis on the proof of the Schoenflies conjecture (and introducing the method of infinite repetition in topology) is 5 pages long.

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According to "Mathematical apocrypha redux" by Krantz, Mazur's thesis was 26 pages long. –  Michael Greinecker Feb 8 '11 at 16:22
Mathscinet says his thesis is 30 pages. –  Jaikrishnan Feb 8 '11 at 16:26
Well, it may not be the shortest but it surelyt appears to have the most variable number of pages! –  Mariano Suárez-Alvarez Feb 8 '11 at 16:42
Let's please heed Timothy's call to do one's homework carefully. "Don't post a candidate unless you're sure your facts are correct, and please give some indication why you're so sure. Read the meta discussion before posting." –  Todd Trimble Feb 8 '11 at 16:47
Excellent, Mariano! –  Georges Elencwajg Feb 8 '11 at 17:20